Noncommutative Christoffel-Darboux Kernels

TL;DR

This paper develops noncommutative Christoffel-Darboux kernels, linking them to classical plurisubharmonic functions and extremal functions, and explores their properties and support notions in free probability.

Contribution

It introduces a new analytic framework for noncommutative kernels, connecting them to classical pluripotential theory and proposing a notion of support for noncommutative distributions.

Findings

Normalized traces of kernels approximate classical plurisubharmonic functions

Kernels are comparable to noncommutative Siciak extremal functions

Estimates for Siciak functions in free product distributions

Abstract

We introduce from an analytic perspective Christoffel-Darboux kernels associated to bounded, tracial noncommutative distributions. We show that properly normalized traces, respectively norms, of evaluations of such kernels on finite dimensional matrices yield classical plurisubharmonic functions as the degree tends to infinity, and show that they are comparable to certain noncommutative versions of the Siciak extremal function. We prove estimates for Siciak functions associated to free products of distributions, and use the classical theory of plurisubharmonic functions in order to propose a notion of support for noncommutative distributions. We conclude with some conjectures and numerical experiments.